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arXiv:1101.2514 · Journal of Physics A Mathematical and Theoretical

On the algebra of local unitary invariants of pure and mixed quantum states

Péter Vrana

Published May 4, 2011• 11 citations
arXivHTMLPDF

Abstract

We study the structure of the inverse limit of the graded algebras of local unitary invariant polynomials using its Hilbert series. For k subsystems, we show that the inverse limit is a free algebra and the number of algebraically independent generators with homogenous degree 2 m equals the number of conjugacy classes of index m subgroups in a free group on k − 1 generators. Similarly, we show that the inverse limit in the case of k -partite mixed state invariants is free and the number of algebraically independent generators with homogenous degree m equals the number of conjugacy classes of index m subgroups in a free group on k generators. The two statements are shown to be equivalent. To illustrate the equivalence, using the representation theory of the unitary groups, we obtain all invariants in the m = 2 graded parts and express them in a simple form both in the case of mixed and pure states. The transformation between the two forms is also derived. Analogous invariants of higher degree are also introduced.

DOI
10.1088/1751-8113/44/22/225304

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We study the structure of the inverse limit of the graded algebras of local unitary invariant polynomials using its Hilbert series. For k subsystems, we show that the inverse limit is a free algebra and the number of algebraically independent generators with homogenous degree 2 m equals the number of conjugacy classes of index m subgroups in a free group on k − 1 generators. Similarly, we show that the inverse limit in the case of k -partite mixed state invariants is free and the number of algebraically independent generators with homogenous degree m equals the number of conjugacy classes of index m subgroups in a free group on k generators. The two statements are shown to be equivalent. To illustrate the equivalence, using the representation theory of the unitary groups, we obtain all invariants in the m = 2 graded parts and express them in a simple form both in the case of mixed and pure states. The transformation between the two forms is also derived. Analogous invariants of higher degree are also introduced.
We show that two density operators of mixed quantum states are in the same local unitary orbit if and only if they agree on polynomial invariants in a certain Noetherian ring for which degree bounds are known in the literature.This implicitly gives a finite complete set of invariants for local unitary equivalence.This is done by showing that local unitary equivalence of density operators is equivalent to local GL equivalence and then using techniques from algebraic geometry and geometric invariant theory.We also classify the SLOCC polynomial invariants and give a degree bound for generators of the invariant ring in the case of n-qubit pure states.Of course it is well known that polynomial invariants are not a complete set of invariants for SLOCC.
We investigate the properties of the inverse limit of the algebras of local unitary invariant polynomials of quantum systems containing various types of fermionic and/or bosonic particles as the dimensions of the single particle state spaces tend to infinity. We show that the resulting algebras are free and present a combinatorial description of an algebraically independent generating set in terms of graphs. These generating sets can be interpreted as minimal sets of polynomial entanglement measures distinguishing between states showing different nonclassical behaviour.
Abstract We calculate the field of rational local unitary invariants for mixed states of two qubits, by employing methods from algebraic geometry. We prove that this field is rational (i.e. purely transcendental), and that it is generated by nine algebraically independent polynomial invariants. We do so by constructing a relative section, in the sense of invariant theory, whose Weyl group is a finite abelian group. From this construction, we are able to give explicit expressions for the generating invariants in terms of the Bloch matrix representation of mixed states of two qubits. We also prove similar rationality results for the local unitary invariants of symmetrically mixed states of two qubits. We also provide a sketch of how to generalize our results to the case of an arbitrary number of qubits. Our results apply to both complex-valued and real-valued invariants.